Advanced Search

We study the one-dimensional motion of the viscous gas represented by the system $v_t-u_x=0$, $u_t+p{\left(v\right)}_x=\mu {(u_x/v)}_x+f\left({\int }_0^{x}v\ddot{x},t\right)$, with the initial and the boundary conditions $(v(x,0),u(x,0))=(v_0\left(x\right),u_0\left(x\right))$, $u(0,t)=u(X,t)=0$. We are concerned with the external forces, namely the function $f$, which do not become small for large time $t$. The main purpose is to show how the solution to this problem behaves around the stationary one, and the proof is based on an elementary $L^2$-energy method.